Normed Spaces, Completeness and Banach Spaces · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

When Cauchy sequences find their limit

Mathematics · Calculus & Analysis · ages 22-24
Name ______________________   Date ____________

  1. Sum of absolute values of 1, negative 2, 3. What is it?

    Answer: ______________

  2. Which properties must a norm satisfy?

    • Positive definiteness, homogeneity, triangle inequality
    • Only positivity
    • Only linearity
    • No properties

  3. Which properties must a length function satisfy to be a norm?

    • Only positivity
    • Only linearity
    • Definiteness, homogeneity, and the triangle inequality

  4. The polynomials with the sup norm on the interval from 0 to 1 form an incomplete space, since the exponential is a missing limit.

    Circle one:   True   False

  5. All norms on a finite dimensional space are equivalent, while in infinite dimensions this can fail.

    Circle one:   True   False

  6. All norms on a finite dimensional space are equivalent, but this fails in infinite dimensions. Is this correct?

    Circle one:   True   False

  7. Which space with sup norm is incomplete and what is the missing limit?

    • Finite dimensional Euclidean space
    • Closed interval
    • Polynomials with sup norm, missing e to the x
    • Finite set

  8. Which normed space is incomplete, and what limit is it missing?

    • Polynomials with the sup norm, missing the exponential
    • Euclidean space of fixed dimension
    • A finite set with any norm

  9. Why do all norms agree in finite dimensions?

    • The unit sphere is compact, so each norm attains positive bounds
    • All finite spaces are automatically incomplete
    • Every norm equals every other norm exactly

  10. Vector entries are double the numbers 1, 2, 3. Largest absolute value is what?

    Answer: ______________

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Answer key

For grown-ups. Fold this page away before handing over the rest.

When Cauchy sequences find their limit W1-mt_y_jueKnovX-s1

  1. 6 · 1 plus 2 plus 3 equals 6.
  2. Positive definiteness, homogeneity, triangle inequality · Positive definiteness, homogeneity, triangle inequality are the three axioms, while the partial lists are incomplete.
  3. Definiteness, homogeneity, and the triangle inequality · All three axioms are needed, and partial lists fall short.
  4. True · Taylor partials are Cauchy yet converge outside the space.
  5. True · Compactness of the sphere gives equivalence, with counterexamples beyond finite dimensions.
  6. True · Compactness of the sphere gives equivalence finitely, with counterexamples infinitely.
  7. Polynomials with sup norm, missing e to the x · Polynomials with sup norm, missing e to the x is incomplete, while Euclidean, closed interval, and finite sets are complete.
  8. Polynomials with the sup norm, missing the exponential · The polynomial space misses the exponential, while the other options are complete.
  9. The unit sphere is compact, so each norm attains positive bounds · Compactness yields the positive minimum and maximum that build the comparison.
  10. 6 · Entries are 2, 4, 6, maximum 6.
Worksheet · LightMySky